text: The simplest case of the instability arises at a planar interface within a porous medium or Hele-Shaw cell, and was treated by Saffman and Taylor but also earlier by other authors. A fluid of viscosity  is driven in the -direction into another fluid of viscosity  at some velocity . Denoting the permeability of the porous medium as a constant, isotropic, , Darcys law gives the unperturbed pressure fields in the two fluids  to bewhere  is the pressure at the planar interface, working in a frame where this interface is instantaneously given by . Perturbing this interface to  (decomposing into normal modes in the  plane, and taking ), the pressure fields becomeAs a consequence of the incompressibility of the flow and Darcys law, the pressure fields must be harmonic, which, coupled with the requirement that the perturbation decay as , fixes  and , with the constants  to be determined by continuity of pressure. Upon linearization, the kinematic boundary condition at the interface (that fluid velocity in the  direction must match the velocity of the fluid interface), coupled with Darcy's law, givesand thus that  and . Matching the pressure fields at the interface givesand so , leading to growth of the perturbation when  - i.e. when the injected fluid is less viscous than the ambient fluid. There are problems with this basic case: namely that the most unstable mode has infinite wavenumber  and grows at an infinitely fast rate, which can be rectified by the introduction of surface tension (which provides a jump condition in pressures across the fluid interface through the Young–Laplace equation), which has the effect of modifying the growth rate to
  with surface tension  and  the mean curvature. This suppresses small-wavelength (high-wavenumber) disturbances, and we would expect to see instabilities with wavenumber  close to the value of  which results in the maximal value of ; in this case with surface tension, there is a unique maximal value.